Half-integer

In mathematics, a half-integer is a number of the form

$n + {1\over 2}$,

where $n$ is an integer. For example,

4½, 7/2, −13/2, 8.5

are all half-integers.

Half-integers occur frequently enough in mathematical contexts that a special term for them is convenient. Note that a half of an integer is not always a half-integer: half of an even integer is an integer but not a half-integer. The half-integers are precisely those numbers that are half of an odd integer, and for this reason are also called the half-odd-integers. Half-integers are a special case of the dyadic rationals, numbers that can be formed by dividing an integer by a power of two.^[1]

Notation and algebraic structure

The set of all half-integers is often denoted

$\mathbb Z + {1\over 2}.$

The integers and half-integers together form a group under the addition operation, which may be denoted^[2]

$\frac{1}{2} \mathbb Z$.

However, these numbers do not form a ring because the product of two half-integers cannot be itself a half-integer.^[3]

Uses

Sphere packing

The densest lattice packing of unit spheres in four dimensions, called the D₄ lattice, places a sphere at every point whose coordinates are either all integers or all half-integers. This packing is closely related to the Hurwitz integers, which are quaternions whose real coefficients are either all integers or all half-integers.^[4]

Physics

In physics, the Pauli exclusion principle results from definition of fermions as particles which have spins that are half-integers.^[5]

The energy levels of the quantum harmonic oscillator occur at half-integers and thus its lowest energy is not zero.^[6]

Sphere volume

Although the factorial function is defined only for integer arguments, it can be extended to fractional arguments using the gamma function. The gamma function for half-integers is an important part of the formula for the volume of an n-dimensional ball of radius R,^[7]

$V_n(R) = \frac{\pi^{n/2}}{\Gamma(\frac{n}{2} + 1)}R^n.$

The values of the gamma function on half-integers are integer multiples of the square root of pi:

$\Gamma\left(\frac{1}{2}+n\right) = \frac{(2n-1)!!}{2^n}\, \sqrt{\pi} = {(2n)! \over 4^n n!} \sqrt{\pi}$

where n!! denotes the double factorial.

References

1. Sabin, Malcolm (2010), Analysis and Design of Univariate Subdivision Schemes, Geometry and Computing 6, Springer, p. 51, ISBN 9783642136481.
2. Turaev, Vladimir G. (2010), Quantum Invariants of Knots and 3-Manifolds, De Gruyter Studies in Mathematics 18 (2nd ed.), Walter de Gruyter, p. 390, ISBN 9783110221848.
3. Boolos, George; Burgess, John P.; Jeffrey, Richard C. (2002), Computability and Logic, Cambridge University Press, p. 105, ISBN 9780521007580.
4. John, Baez (August 12, 2004), "On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry by John H. Conway and Derek A. Smith", Bulletin of the American Mathematical Society 42: 229–243, doi:10.1090/S0273-0979-05-01043-8 Check date values in: |year= / |date= mismatch (help).
5. Mészáros, Péter (2010), The High Energy Universe: Ultra-High Energy Events in Astrophysics and Cosmology, Cambridge University Press, p. 13, ISBN 9781139490726.
6. Fox, Mark (2006), Quantum Optics : An Introduction, Oxford Master Series in Physics 6, Oxford University Press, p. 131, ISBN 9780191524257.
7. Equation 5.19.4, NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.6 of 2013-05-06.